\xiti
\begin{xiaotis}

\xiaoti{求下列各式的值：}
\begin{xiaoxiaotis}

    \begin{tabular}[t]{*{2}{@{}p{18em}}}
        \xiaoxiaoti{$\sin45^\circ \cos15^\circ$；} & \xiaoxiaoti{$\cos75^\circ \cos15^\circ$；} \\
        \xiaoxiaoti{$\cos157^\circ30' \sin22^\circ30'$；} & \xiaoxiaoti{$\cos20^\circ \cos40^\circ \cos80^\circ$；} \\
        \xiaoxiaoti{$\sin20^\circ \sin40^\circ \sin80^\circ$；} & \xiaoxiaoti{$\tan10^\circ \tan50^\circ \tan70^\circ$。}
    \end{tabular}

\end{xiaoxiaotis}

\xiaoti{把下列各式化为和或差的形式：}
\begin{xiaoxiaotis}

    \renewcommand\arraystretch{1.5}
    \begin{tabular}[t]{*{2}{@{}p{18em}}}
        \xiaoxiaoti{$2\sin\dfrac{3\alpha}{2} \cos\dfrac{\alpha}{2}$；} & \xiaoxiaoti{$\sin ax \sin bx$；} \\
        \xiaoxiaoti{$\cos(\alpha + \beta) \cos(\alpha - \beta)$；} & \xiaoxiaoti{$\cos\left( \dfrac{3\pi}{4} + 2\alpha \right) \sin\left( \dfrac{\pi}{4} - \alpha \right)$；} \\
        \xiaoxiaoti{$\sin x \cos 3x$。}
    \end{tabular}

\end{xiaoxiaotis}

\xiaoti{把下列各式化为积的形式：}
\begin{xiaoxiaotis}

    \renewcommand\arraystretch{1.5}
    \begin{tabular}[t]{*{2}{@{}p{18em}}}
        \xiaoxiaoti{$\sin28^\circ + \cos17^\circ$；} & \xiaoxiaoti{$\cos54^\circ - \sin54^\circ$；} \\
        \xiaoxiaoti{$\cos\left( x - \dfrac{\pi}{4} \right) - \cos\left( x + \dfrac{\pi}{4} \right)$；} & \xiaoxiaoti{$\cos\left( \dfrac{\pi}{6} + \alpha \right) + \cos\left( \dfrac{\pi}{6} - \alpha \right)$；} \\
        \xiaoxiaoti{$\dfrac{\sqrt{2}}{2} + \cos\alpha$；} & \xiaoxiaoti{$1 + \sin 2A$；} \\
        \xiaoxiaoti{$1 + \sqrt{3}\tan\alpha$；} & \xiaoxiaoti{$3 - 4\sin^2\alpha$；} \\
        \xiaoxiaoti{$\sin^2\theta - \sin^2\varphi$；} & \xiaoxiaoti{$\cos^2x - \cos^2y$。}
    \end{tabular}

\end{xiaoxiaotis}

\xiaoti{证明下列各恒等式：}
\begin{xiaoxiaotis}

    \jiange
    \xiaoxiaoti{$\dfrac{\sin A + \sin 3A + \sin 5A}{\sin 3A + \sin 5A + \sin 7A} = \dfrac{\sin 3A}{\sin 5A}$；}\jiange

    \xiaoxiaoti{$\dfrac{\sin\alpha + \sin3\alpha + \sin5\alpha}{\cos\alpha + \cos3\alpha + \cos5\alpha} = \tan3\alpha$；}\jiange

    \xiaoxiaoti{$\dfrac{\sin x - \sin y}{\sin(x + y)} = \dfrac{\sin\dfrac{1}{2}(x - y)}{\sin\dfrac{1}{2}(x + y)}$；}\jiange

    \xiaoxiaoti{$\dfrac{\cos2\alpha + \cos2\beta}{1 + \cos2(\alpha + \beta)} = \dfrac{\cos(\alpha - \beta)}{\cos(\alpha + \beta)}$；}\jiange

    \xiaoxiaoti{$\sec\left( \dfrac{\pi}{4} + \alpha \right) \sec\left( \dfrac{\pi}{4} - \alpha \right) = 2\sec2\alpha$；}\jiange

    \xiaoxiaoti{$\cos^2A + \cos^2(60^\circ - A) + \cos^2(60^\circ + A) = \dfrac{3}{2}$；}\jiange

    \xiaoxiaoti{$\sin\dfrac{\alpha}{2} \sin\dfrac{7\alpha}{2} + \sin\dfrac{3\alpha}{2} \sin\dfrac{11\alpha}{2} = \sin2\alpha \sin5\alpha$；}\jiange

    \xiaoxiaoti{$(\cos x + \sin x)(\cos2x + \sin2x) = \cos x + \sin3x$。}

\end{xiaoxiaotis}

\xiaoti{求下列各式的值：}
\begin{xiaoxiaotis}

    \xiaoxiaoti{$\cos20^\circ + \cos100^\circ + \cos140^\circ$；}\jiange

    \xiaoxiaoti{$\dfrac{\cos80^\circ - \cos20^\circ}{\sin80^\circ + \sin20^\circ}$；}\jiange

    \xiaoxiaoti{$\cos40^\circ + \cos60^\circ + \cos80^\circ + \cos160^\circ$；}

    \xiaoxiaoti{$\cos40^\circ \cos80^\circ + \cos80^\circ \cos160^\circ + \cos160^\circ \cos40^\circ$；}

    \xiaoxiaoti{$\sin20^\circ \cdot \sin40^\circ \cdot \sin60^\circ \cdot \sin80^\circ$。}

\end{xiaoxiaotis}

\xiaoti{在 $\triangle ABC$ 中，求证：}
\begin{xiaoxiaotis}

    \xiaoxiaoti{$\sin A + \sin B - \sin C = 4\sin\dfrac{A}{2} \sin\dfrac{B}{2} \cos\dfrac{C}{2}$；}\jiange

    \xiaoxiaoti{$\cos A + \cos B + \cos C = 1 + 4\sin\dfrac{A}{2} \sin\dfrac{B}{2} \sin\dfrac{C}{2}$；}\jiange

    \xiaoxiaoti{$\sin 2A + \sin 2B + \sin 2C = 4\sin A \sin B \sin C$；}

    \xiaoxiaoti{$\cos 2A + \cos 2B + \cos 2C = -1 -4\cos A \cos B \cos C$。}

\end{xiaoxiaotis}

\xiaoti{将下列各式化为一个角的一个三角函数的形式：}
\begin{xiaoxiaotis}

    \begin{tabular}[t]{*{2}{@{}p{18em}}}
        \xiaoxiaoti{$3\cos\alpha - 4\sin\alpha$；} & \xiaoxiaoti{$5\sin\varphi + 12\cos\varphi$；} \\
        \xiaoxiaoti{$4\sin t + 3\cos t$；} & \xiaoxiaoti{$7\sin 2t - 6\cos 2t$。}
    \end{tabular}

\end{xiaoxiaotis}

\xiaoti{求下列各式的最大值和最小值：}
\begin{xiaoxiaotis}

    \renewcommand\arraystretch{1.5}
    \begin{tabular}[t]{*{2}{@{}p{18em}}}
        \xiaoxiaoti{$\sin x \cos x$；} & \xiaoxiaoti{$\sin\left( \alpha + \dfrac{\pi}{4} \right) + \sin\left( \alpha - \dfrac{\pi}{4} \right)$；} \\
        \xiaoxiaoti{$\cos\left( \dfrac{\pi}{3} + 2\theta \right) \sin\left( \dfrac{\pi}{3} - 2\theta \right)$；} & \xiaoxiaoti{$6\cos\theta + 8\sin\theta$。}
    \end{tabular}

\end{xiaoxiaotis}


\end{xiaotis}